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Solving Equations by Substitution

Solving a system of equations means to solve for a numerical value for unknown variables. Substitution is a special method wherein we substitute the value of one variable into an equation to get the value of other unknown variable.

Substitution Method: Substitution is one of the methods which is used to solve a system of equation, which requires the following two important points to be satisfied:

1)  There should be almost the similar number of equations as there would be unknown variables present in the system of equations.

2)  Any one of the equations from the given two systems could be easily solved for one variable.

If the above two conditions are satisfied, then only we can use substitution as a method for solving a system of equations.

If there are two variables, there must be two equations; three variables, three equations, and so on.
One of the equations from the given two systems could be easily solved for one variable is the main idea of substitution method.

Steps for Solving Equations by Substitution:

To solve a system of equations with two variables we require the following steps:

Step 1: Choose any one equation that is easy to solve and make independent that one unknown variable; this equation will be considered as the first equation.

Step 2: Substitute the solution got from solving the above equation into the second equation and solve for the variable in the changed equation.

Step 3: Using the value of the first unknown variable, got from the second equation, we will now substitute it into the first equation and solve for the second unknown variable.

Step 4: Now we will substitute the values for both variables into both the equations to prove that they are correct and valid.

Solved Examples:

Example 1: Solve the following system by substitution. 2 x – 3 y = –2 and 4x + y  = 24.
Solution: In substitution method, we will solve one of the equations for one of the variables, and putting that value in the other equation. Here, we choose 4x + y = 24, it can be rewritten as:
y = –4x + 24, now we will substitute it for "y" in the first equation, and solve for x, that is, 2x – 3 (–4x + 24) = –2 or 2x + 12x – 72 = –2 or 14x = 70, this implies:
x = 5.
Now y = –4(5) + 24 = –20 + 24 = 4
Hence, the solution is (x, y) = (5, 4).

Example 2: Solve x + y = 20 and x − y = 10 by substitution method.
Solution: Consider, x − y = 10 this implies, x = 10 + y
Putting it in, x + y = 20, we will get 10 + y + y = 20 or 10 + 2y = 20, that is,
2y = 10 or y = 5.
Replacing y into x + y = 20 gives x + 5 = 20, that is x = 15.
Hence, the solution to the system is x = 15 and y = 5.

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